HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem sinperlem2 8625
Description: Lemma for sin2kpi 8626 and cos2kpi 8627.
Assertion
Ref Expression
sinperlem2 |- (K e. ZZ -> ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0))
Proof of Theorem sinperlem2
StepHypRef Expression
1 elznn0 6104 . . . . 5 |- (K e. ZZ <-> (K e. RR /\ (K e. NN0 \/ -uK e. NN0)))
21pm3.27bi 326 . . . 4 |- (K e. ZZ -> (K e. NN0 \/ -uK e. NN0))
3 sinperlem1 8624 . . . . 5 |- (K e. NN0 -> ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0))
4 sinperlem1 8624 . . . . 5 |- (-uK e. NN0 -> ((cos` (-uK x. (2 x. pi))) = 1 /\ (sin` (-uK x. (2 x. pi))) = 0))
53, 4orim12i 336 . . . 4 |- ((K e. NN0 \/ -uK e. NN0) -> (((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) \/ ((cos` (-uK x. (2 x. pi))) = 1 /\ (sin` (-uK x. (2 x. pi))) = 0)))
62, 5syl 10 . . 3 |- (K e. ZZ -> (((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) \/ ((cos` (-uK x. (2 x. pi))) = 1 /\ (sin` (-uK x. (2 x. pi))) = 0)))
7 mulneg1t 5431 . . . . . . . . 9 |- ((K e. CC /\ (2 x. pi) e. CC) -> (-uK x. (2 x. pi)) = -u(K x. (2 x. pi)))
8 zcnt 6095 . . . . . . . . 9 |- (K e. ZZ -> K e. CC)
9 2cn 5935 . . . . . . . . . . 11 |- 2 e. CC
10 pire 8615 . . . . . . . . . . . 12 |- pi e. RR
1110recn 5294 . . . . . . . . . . 11 |- pi e. CC
129, 11mulcl 5301 . . . . . . . . . 10 |- (2 x. pi) e. CC
1312a1i 8 . . . . . . . . 9 |- (K e. ZZ -> (2 x. pi) e. CC)
147, 8, 13sylanc 471 . . . . . . . 8 |- (K e. ZZ -> (-uK x. (2 x. pi)) = -u(K x. (2 x. pi)))
1514fveq2d 3719 . . . . . . 7 |- (K e. ZZ -> (cos` (-uK x. (2 x. pi))) = (cos` -u(K x. (2 x. pi))))
1615eqeq1d 1480 . . . . . 6 |- (K e. ZZ -> ((cos` (-uK x. (2 x. pi))) = 1 <-> (cos` -u(K x. (2 x. pi))) = 1))
17 axmulcl 5253 . . . . . . . . 9 |- ((K e. CC /\ (2 x. pi) e. CC) -> (K x. (2 x. pi)) e. CC)
1817, 8, 13sylanc 471 . . . . . . . 8 |- (K e. ZZ -> (K x. (2 x. pi)) e. CC)
19 cosnegt 7393 . . . . . . . 8 |- ((K x. (2 x. pi)) e. CC -> (cos` -u(K x. (2 x. pi))) = (cos` (K x. (2 x. pi))))
2018, 19syl 10 . . . . . . 7 |- (K e. ZZ -> (cos` -u(K x. (2 x. pi))) = (cos` (K x. (2 x. pi))))
2120eqeq1d 1480 . . . . . 6 |- (K e. ZZ -> ((cos` -u(K x. (2 x. pi))) = 1 <-> (cos` (K x. (2 x. pi))) = 1))
2216, 21bitr2d 528 . . . . 5 |- (K e. ZZ -> ((cos` (K x. (2 x. pi))) = 1 <-> (cos` (-uK x. (2 x. pi))) = 1))
23 sinnegt 7392 . . . . . . . 8 |- ((K x. (2 x. pi)) e. CC -> (sin` -u(K x. (2 x. pi))) = -u(sin` (K x. (2 x. pi))))
2418, 23syl 10 . . . . . . 7 |- (K e. ZZ -> (sin` -u(K x. (2 x. pi))) = -u(sin` (K x. (2 x. pi))))
2524eqeq1d 1480 . . . . . 6 |- (K e. ZZ -> ((sin` -u(K x. (2 x. pi))) = 0 <-> -u(sin` (K x. (2 x. pi))) = 0))
2614fveq2d 3719 . . . . . . 7 |- (K e. ZZ -> (sin` (-uK x. (2 x. pi))) = (sin` -u(K x. (2 x. pi))))
2726eqeq1d 1480 . . . . . 6 |- (K e. ZZ -> ((sin` (-uK x. (2 x. pi))) = 0 <-> (sin` -u(K x. (2 x. pi))) = 0))
28 sinclt 7381 . . . . . . 7 |- ((K x. (2 x. pi)) e. CC -> (sin` (K x. (2 x. pi))) e. CC)
29 negeq0t 5770 . . . . . . 7 |- ((sin` (K x. (2 x. pi))) e. CC -> ((sin` (K x. (2 x. pi))) = 0 <-> -u(sin` (K x. (2 x. pi))) = 0))
3018, 28, 293syl 20 . . . . . 6 |- (K e. ZZ -> ((sin` (K x. (2 x. pi))) = 0 <-> -u(sin` (K x. (2 x. pi))) = 0))
3125, 27, 303bitr4rd 550 . . . . 5 |- (K e. ZZ -> ((sin` (K x. (2 x. pi))) = 0 <-> (sin` (-uK x. (2 x. pi))) = 0))
3222, 31anbi12d 627 . . . 4 |- (K e. ZZ -> (((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) <-> ((cos` (-uK x. (2 x. pi))) = 1 /\ (sin` (-uK x. (2 x. pi))) = 0)))
3332orbi2d 613 . . 3 |- (K e. ZZ -> ((((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) \/ ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0)) <-> (((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) \/ ((cos` (-uK x. (2 x. pi))) = 1 /\ (sin` (-uK x. (2 x. pi))) = 0))))
346, 33mpbird 196 . 2 |- (K e. ZZ -> (((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) \/ ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0)))
35 oridm 243 . 2 |- ((((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0) \/ ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0)) <-> ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0))
3634, 35sylib 198 1 |- (K e. ZZ -> ((cos` (K x. (2 x. pi))) = 1 /\ (sin` (K x. (2 x. pi))) = 0))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956  ` cfv 3177  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214  1c1 5215   x. cmul 5219  -ucneg 5273  NN0cn0 5277  ZZcz 5278  2c2 5916  sincsin 7245  cosccos 7246  picpi 7247
This theorem is referenced by:  sin2kpi 8626  cos2kpi 8627
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-iin 2564  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-map 4314  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-r1 4623  df-rank 4624  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-2 5925  df-3 5926  df-4 5927  df-5 5928  df-6 5929  df-7 5930  df-8 5931  df-9 5932  df-n0 6055  df-z 6091  df-fl 6180  df-q 6202  df-rp 6227  df-seq1 6253  df-shft 6286  df-ioo 6306  df-ioc 6307  df-ico 6308  df-icc 6309  df-uz 6358  df-fz 6408  df-seqz 6473  df-seq0 6474  df-exp 6509  df-sqr 6608  df-re 6690  df-im 6691  df-cj 6692  df-abs 6693  df-fac 6877  df-bc 6902  df-clim 6921  df-sum 6926  df-cncf 7206  df-ef 7248  df-sin 7250  df-cos 7251  df-pi 7252  df-top 7542  df-cn 7704  df-cnp 7705  df-met 7743  df-bl 7745  df-opn 7746  df-lm 7874
Copyright terms: Public domain